PlanetMath: theorems on complex function series

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theorems on complex function series

(Theorem)

Theorem 1. If the complex functions $f_1,\,f_2,\,f_3,\,\ldots$ are continuous on the path $\gamma$ and the series

$\displaystyle f_1(z)+f_2(z)+f_3(z)+\cdots$

(1)

converges uniformly on $\gamma$ to the sum function

, then one has

$\displaystyle \int_{\gamma}F(z)\,dz = \int_{\gamma}f_1(z)\,dz+\int_{\gamma}f_2(z)\,dz+\int_{\gamma}f_3(z)\,dz+\cdots$

Theorem 2. If the functions $f_1,\,f_2,\,f_3,\,\ldots$ are holomorphic in a domain and the series (1) converges uniformly in every closed disc of , then also the sum function of (1) is holomorphic in and

$\displaystyle \frac{d^nF(z)}{dz^n} = F^{(n)}(z) = f_1^{(n)}(z)+f_2^{(n)}(z)+f_3^{(n)}(z)+\cdots$

(2)

is true for every positive integer

in all points of

. The series (2) converges uniformly in every compact subdomain of

Theorem 3. If is holomorphic in a domain and is a point of , then one can expand to a power series (the so-called Taylor series)

$\displaystyle f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n \quad \mathrm{where} \quad a_n = \frac{f^{(n)}(z_0)}{n!}\quad(n = 0,\,1,\,2,\,\ldots).$

This expansion is valid at least in the greatest disk $\vert z-z_0\vert < r\, (\leqq \infty)$ which contains points of

only.

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See Also: identity theorem of power series, Weierstrass double series theorem

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Cross-references: contains, Taylor series, power series, expand, compact, points, integer, positive, disc, domain, holomorphic, functions, sum function, converges uniformly, series, path, continuous, complex functions
There are 4 references to this entry.

This is version 5 of theorems on complex function series, born on 2007-03-05, modified 2008-05-01.
Object id is 9032, canonical name is TheoremsOnComplexFunctionSeries.
Accessed 667 times total.

Classification:

AMS MSC:	40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)
	30B99 (Functions of a complex variable :: Series expansions :: Miscellaneous)

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